Nonequilibrium protein complexes as molecular automata

Abstract

Biology stores information and computes at the molecular scale, yet the ways in which it does so are often distinct from human-engineered computers. Mapping biological computation onto architectures familiar to computer science remains an outstanding challenge. Here, inspired by Crick's proposal for molecular memory, we analyse a thermodynamically-consistent model of a protein complex subject to driven, nonequilibrium enzymatic reactions. In the strongly driven limit, we find that the system maps onto a stochastic, asynchronous variant of cellular automata, where each rule corresponds to a different set of enzymes being present. We find a broad class of phenomena in these 'molecular automata' that can be exploited for molecular computation, including error-tolerant memory via multistable attractors, and long transients that can be used as molecular stopwatches. By systematically enumerating all possible dynamical rules, we identify those that allow molecular automata to implement simple computational architectures such as finite-state machines. Overall, our results provide a framework for engineering synthetic molecular automata, and offer a route to building protein-based computation in living cells.

Figures (6)

• Figure 1: Nonequilibrium transitions in a protein complex realise cellular automata rules. (a) Schematic of a protein complex made of identical monomers, each of which can be post-translationally modified (e.g., phosphorylated). We represent the state of the complex by a binary string, which indicates the modification state of each monomer. (b) Transitions are catalysed by enzymes that add or remove modifications; enzyme recruitment to a site depends on the modification status of its neighbours. (c) The eight possible enzymes can each be represented by a binary triplet, denoting the state of the monomer triplet to which they preferentially bind (substrate) before the action of the enzyme. The set of enzymes present in the system, and therefore the 'rule' (set of allowed transitions), is captured in the Wolfram code which assigns rules a number between 0 and 255. This example with all eight enzymes present corresponds to rule 51. (d) Another example, with only two enzymes present, corresponding to rule 142. (e) Simulated stochastic trajectories of a protein complex of size $N = 6$ under different rules, showing a nonequilibrium travelling wave (rule 142), localization to one of two single state attractors (rule 170), and equilibrium-like switching between a subset of states (rule 124).
• Figure 2: Conformations are funnelled into distinct stable attractors. Shown are the number of states in attractors for each of the 88 non-redundant rule-sets (labelled by their Wolfram codes), for a complex of size $N=6$. States in the same attractor are grouped together into boxes, which are coloured to denote the type of attractor. Rules explicitly discussed in Figs. \ref{['fig:1']}--\ref{['fig:4']} have been highlighted. The insets show three rule-sets as directed graphs, where nodes denote a unique state of the complex, and transitions are catalysed reactions between states. States not in an attractor are in gray, others are coloured.
• Figure 3: Some rules can be highly sensitive to the size of the complex, while others remain robust. Transition graphs for rules 124 and 6, for complex sizes $N=5$ and $N=6$. Colours are as in Fig. \ref{['fig:2']}. The attractor structure of rule 124 is robust, showing an equilibrium multistate attractor and a single-state attractor in both cases. Rule 6 is highly sensitive to whether the complex size is odd or even, showing a nonequilibrium cycle coexisting with a single-state attractor for $N=5$, and just three single-state attractors for $N=6$.
• Figure 4: Relaxation kinetics and molecular stopwatches (a) Rank-ordered plot of inverse spectral gaps, i.e. longest relaxation times $\tau_\lambda$, across all 88 rules for an octamer with $N=8$. Inset: five replicas of stochastic simulations for rule 166, which has the longest relaxation time. (b) Relaxation time as a function of complex size $N$ for rule 166, showing exponential scaling. (c) Distribution of arrival times to the attractor $\mathbf{1}$ for a population of complexes with $N=8$ initialized in a random state, for rule 166. Inset: the fraction of complexes that have not reached the attractor scales as $e^{-t/\tau_\lambda}$.
• Figure 5: Error correction under finite driving and spontaneous reactions. (a) Splitting probability towards the $\textbf{0}$ and $\textbf{1}$ attractors, as well as towards any other attractor, for rules 232 and 170, as a function of the number of ones in the complex. Rule 232 demonstrates perfect error correction under single bit flip errors, with a splitting probability of 1 away from $\textbf{0}$ or $\textbf{1}$. (b) Coherence time $\tau_{10}$, representing how long it takes $10\%$ of complexes to exit the $\textbf{0}$ state, for both rules as a function of the nonequilibrium driving $\Delta \mu$, for several values of the spontaneous transition rate $k_\mathrm{spo}$ (in units of $k_\mathrm{cat}$).